Bosonic code

This code defines the Bosonic Kingdom


Also called an oscillator or a continuous-variable (CV) code. Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode). States of a single oscillator are elements of the Hilbert space of \(\ell^2\)-normalizable functions on \(\mathbb{R}\). Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.


An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes. For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). For multiple modes, error set elements are tensor products of elements of the single-qudit error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for trace-class linear operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product [1]. There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code.

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the amplitude damping (a.k.a. photon loss or attenuation) noise channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set. A definition of distance associated with this error set is the minimum weight of a loss error that implements a nontrivial logical operation in the code.

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) [2] and its adjoint (a.k.a. phasors [3]) along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These also allow one to expand trace-class operators, but do not form an orthonormal set [4]. These operators are correspong to a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements.


The quantum capacity of the pure-loss channel [5] and the dephasing noise channel [6] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [7][8].


Reviews on bosonic codes can be found in Refs. [4][9][10][11][12][13].For an introduction to continuous-variable quantum systems, see the books [14][15].




  • Bosonic c-q code — Bosonic c-q codes are bosonic codes designed to transmit classical information.
  • Fermionic code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.


K. E. Cahill and R. J. Glauber, “Ordered Expansions in Boson Amplitude Operators”, Physical Review 177, 1857 (1969) DOI
L. Susskind and J. Glogower, “Quantum mechanical phase and time operator”, Physics Physique физика 1, 49 (1964) DOI
J. Bergou and B.-G. Englert, “Operators of the phase. Fundamentals”, Annals of Physics 209, 479 (1991) DOI
V. V. Albert, “Bosonic coding: introduction and use cases”, (2022) arXiv:2211.05714
M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
L. Lami and M. M. Wilde, “Exact solution for the quantum and private capacities of bosonic dephasing channels”, (2022) arXiv:2205.05736
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020) arXiv:2002.11008 DOI
A. Joshi, K. Noh, and Y. Y. Gao, “Quantum information processing with bosonic qubits in circuit QED”, Quantum Science and Technology 6, 033001 (2021) arXiv:2008.13471 DOI
W. Cai et al., “Bosonic quantum error correction codes in superconducting quantum circuits”, Fundamental Research 1, 50 (2021) arXiv:2010.08699 DOI
K. Noh, “Quantum Computation and Communication in Bosonic Systems”, (2021) arXiv:2103.09445
S. M. Girvin, “Introduction to Quantum Error Correction and Fault Tolerance”, (2023) arXiv:2111.08894
N. J. Cerf, G. Leuchs, and E. S. Polzik, Quantum Information with Continuous Variables of Atoms and Light (PUBLISHED BY IMPERIAL COLLEGE PRESS AND DISTRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO., 2007) DOI
A. Serafini, “Quantum Continuous Variables”, (2017) DOI
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Zoo Code ID: oscillators

Cite as:
“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_oscillators, title={Bosonic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.